Low energy effective action of domain-wall fermion and the Ginsparg-Wilson relation

We derive the effective action of the light fermion field of the domain-wall fermion, which is referred as $q(x)$ by Furman and Shamir. The inverse of the effective Dirac operator turns out to be identical to the inverse of the truncated overlap Dirac operator, except a local contact term which would give the chiral symmetry breaking in the Ginsparg-Wilson relation. This result allows us to relate the light fermion field and the fermion field described by the truncated overlap Dirac operator and to understand the chiral property of the light fermion through the exact chiral symmetry based on the Ginsparg-Wilson relation.